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In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space.
Once a local basis is chosen, the metric tensor
appears as a matrix, conventionally notated as G (see also
metric). The notation gij is conventionally used for the
components of the metric tensor. (i.e. the elements of the matrix.) (In the following, we use the Einstein summation convention.)
The length of a segment of a curve parameterized by t, from a to b, is defined as:
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The angle between two tangent vectors, U and V, is defined
as:
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To compute the metric tensor from a set of equations relating the space to cartesian space (gij = δij: see Kronecker delta for more details), compute the jacobian
of the set of equations, and multiply (outer product) the transpose of that jacobian by the jacobian.
- G = JTJ
Example
Given a two-dimensional Euclidean metric tensor:
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The length of a curve reduces to the familiar Calculus formula:
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Some basic Euclidean metrics
Polar coordinates: (x1,x2) = (r,θ)
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Cylindrical coordinates: (x1,x2,x3) =
(r,θ,z)
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Spherical coordinates: (x1,x2,x3) =
(r,φ,θ)
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